I have the following undirected graph (picture) that contains a cycle or a Hamiltonian path of length |V|= 8. The cycle (path) with no repeated edges and vertices is the red line. The adjacency matrix is :

How can I plot this graph in R ?

Ham = matrix(c(0,1,0,1,1,0,0,0,
             1,0,1,0,0,1,0,0,
             0,1,0,1,0,0,0,1,
             1,0,1,0,0,0,1,0,
             1,0,0,0,0,1,1,0,
             0,1,0,0,1,0,0,1,
             0,0,0,1,1,0,0,1,
             0,0,1,0,0,1,1,0),8,8)
Ham

Update

If you need only one of all the Hamilton circles, you can try graph.subisomorphic.lad (thanks for the advice from @Szabolcs), which speeds up a lot if you don't need to list out all the possibilities, e.g.,

g <- graph_from_adjacency_matrix(Ham, "undirected")
es <- graph.subisomorphic.lad(make_ring(vcount(g)), g)$map
g %>%
  set_edge_attr("color", value = "black") %>%
  set_edge_attr("color",
    get.edge.ids(g, c(rbind(es, c(es[-1], es[1])))),
    value = "red"
  ) %>%
  plot()

If you want to find all Hamilton circles:

You should be aware of the fact that the Hamilton circle is isomorphic to a ring consisting of all vertices, so we can resort to subgraph_isomorphisms to find out all those kinds of "rings", e.g.,

g <- graph_from_adjacency_matrix(Ham, "undirected")
lst <- lapply(
  subgraph_isomorphisms(make_ring(vcount(g)), g),
  function(es) {
    g %>%
      set_edge_attr("color", value = "black") %>%
      set_edge_attr("color",
        get.edge.ids(g, c(rbind(es, c(es[-1], es[1])))),
        value = "red"
      )
  }
)

where lst is a list of graphs, and you can see

1. plot(lst[[1]] gives
2. plot(lst[[2]] gives

and so on so forth.